We investigate native point defects and rare-earth (co)dopants in monoclinic $\mathrm{Sr}\mathrm{Al}$${}_{2}$$\mathrm{O}$${}_{4}$ using hybrid density-functional defect calculations. Europium ($\mathrm{Eu}$) and dysprosium ($\mathrm{Dy}$) are found to be mixed valence and energetically most favorable at the $\mathrm{Sr}$ lattice sites. However, unlike $\mathrm{Eu}$, where both $\mathrm{Eu}$${}^{2+}$ and $\mathrm{Eu}$${}^{3+}$ can be realized in synthesis, $\mathrm{Dy}$ is stable predominantly as $\mathrm{Dy}$${}^{3+}$ and the divalent $\mathrm{Dy}$${}^{2+}$ may only be photogenerated under irradiation. On the basis of an analysis of $\mathrm{Eu}$-related band--defect (including charge-transfer) and interconfigurational $5d$--$4f$ optical transitions, we assign the characteristic broad blue (445-nm) and green (520-nm) emission bands in $\mathrm{Eu}$${}^{2+}$-doped $\mathrm{Sr}\mathrm{Al}$${}_{2}$$\mathrm{O}$${}_{4}$ to the $4{f}^{6}5{d}^{1}\ensuremath{\rightarrow}4{f}^{7}$ transition in $\mathrm{Eu}$${}^{2+}$ incorporated at the $\mathrm{Sr}$1 and $\mathrm{Sr}$2 sites, respectively. Strontium interstitials (${\mathrm{Sr}}_{i}$; not oxygen vacancies, in contrast to what is commonly believed) and substitutional Dy impurities (${\mathrm{Dy}}_{\mathrm{Sr}}$) can act as efficient electron traps for room-temperature persistent luminescence. This work calls for a reassessment of certain assumptions regarding specific carrier trapping centers made in all mechanisms previously proposed for the persistent luminescence in $\mathrm{Eu}$- and ($\mathrm{Eu}$,$\mathrm{Dy}$)-doped $\mathrm{Sr}\mathrm{Al}$${}_{2}$$\mathrm{O}$${}_{4}$. It also serves as a methodological template for the understanding and design of rare-earth doped phosphors.